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On minimally Free Algebras

Published online by Cambridge University Press:  20 November 2018

Paul Bankston  and
Richard Schutt
Paul Bankston
Affiliation:
Marquette University, Milwaukee, Wisconsin
Richard Schutt
Affiliation:
2546 North 7th Street, Milwaukee, Wisconsin
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For us an “algebra” is a finitary “universal algebra” in the sense of G. Birkhoff [9]. We are concerned in this paper with algebras whose endomorphisms are determined by small subsets. For example, an algebra A is rigid (in the strong sense) if the only endomorphism on A is the identity idA. In this case, the empty set determines the endomorphism set E(A). We place the property of rigidity at the bottom rung of a cardinal-indexed ladder of properties as follows. Given a cardinal number κ, an algebra A is minimally free over a set of cardinality κ (κ-free for short) if there is a subset X ⊂ A of cardinality κ such that every function f:X → A extends to a unique endomorphism ϕ ∊ E(A). (It is clear that A is rigid if and only if A is 0-free.) Members of X will be called counters; and we will be interested in how badly counters can fail to generate the algebra.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 5 , 01 October 1985 , pp. 963 - 978
Copyright
Copyright © Canadian Mathematical Society 1985

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